Question

The distance between the Earth and the moon is approximately $384000\ km$. Explain this distance in meters in exponential notation.

Answer 1

Hint: First of all we will convert km into meter by unit by using $1km=1000m$and then we will convert rest of the zeroes into exponential form and then we will add the exponents of two values by using the formula of ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$ and we will find the answer.

Complete step by step answer:
Now, in question it is given that the distance between Earth and moon is approximately $384000\ km$. So, first of all we will convert $384000\ km$ into exponential form which can be given mathematically as,
$384000\ km=3.84\times {{10}^{5}}km$
Here, it can be said that by shifting the zeroes by 5 decimals we can keep that 5 decimals in power to the 10.
Now, we will convert kilometer into meter by using the fact that $1km=1000m$
So, the expression can be written as,
$384000\ km=3.84\times {{10}^{5}}\times 1000m$
Again, shifting the zeroes up to 3 decimals and keeping that decimals in power of 10 we will get,
$384000\ km=3.84\times {{10}^{5}}\times {{10}^{3}}m$
Now, as the base is same, we can add the exponents of 10, so, we will use the rule of addition of exponents which can be given as,
${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$
Here, $a=10$,$m=5$, and $n=3$, on substituting these values we will get,
${{10}^{5}}\times {{10}^{3}}={{10}^{5+3}}={{10}^{8}}$
So, replacing it in the expression we will get,
$384000\ km=3.84\times {{10}^{8}}m$.
Thus, the value of distance between the Earth and the moon can be given in meters in exponential form as $3.84\times {{10}^{8}}m$ .

Note: Here, the base 10 to the exponents was same so we could apply the addition rule of exponents, if the base is not same for example, ${{10}^{5}}\times {{12}^{3}}$ then, ${{10}^{5}}\times {{12}^{3}}\ne {{10}^{5+3}}$, which means we cannot apply that rule in such cases. Student might make mistake in using the addition rule, they might multiply the exponents such as ${{10}^{5}}\times {{10}^{3}}={{10}^{5\times 3}}={{10}^{15}}$ instead of, ${{10}^{5}}\times {{10}^{3}}={{10}^{5+3}}$, but this is wrong again and the answer will also be wrong, so, students must use all the rules carefully.